Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/464

Rh magneto-electric induction. In this he was eminently successful, and we shall indicate the method by which the laws of induced currents may be deduced from Weber s formula. But we must observe, that the circumstance that a law deduced from the pheno mena discovered by Ampère is able also to account for the phenomena afterwards discovered by Faraday does not give so much additional weight to the evidence for the physical truth of the law as we might at first suppose.

For it has been shewn by Helmholtz and Thomson (see Art. 543), that if the phenomena of Ampère are true, and if the principle of the conservation of energy is admitted, then the phenomena of induction discovered by Faraday follow of necessity. Now Weber's law, with the various assumptions about the nature of electric currents which it involves, leads by mathematical transformations to the formula of Ampère. Weber's law is also consistent with the principle of the conservation of energy in so far that a potential exists, and this is all that is required for the application of the principle by Helmholtz and Thomson. Hence we may assert, even before making any calculations on the subject, that Weber's law will explain the induction of electric currents. The fact, therefore, that it is found by calculation to explain the induction of currents, leaves the evidence for the physical truth of the law exactly where it was.

On the other hand, the formula of Gauss, though it explains the phenomena of the attraction of currents, is inconsistent with the principle of the conservation of energy, and therefore we cannot assert that it will explain all the phenomena of induction. In fact, it fails to do so, as we shall see in Art. 859.

857.] We must now consider the electromotive force tending to produce a current in the element $$ds'$$ due to the current in $$ds$$, when $$ds$$ is in motion, and when the current in it is variable.

According to Weber, the action on the material of the conductor of which $$ds'$$ is an element, is the sum of all the actions on the electricity which it carries. The electromotive force, on the other hand, on the electricity in $$ds'$$ is the difference of the electric forces acting on the positive and the negative electricity within it. Since all these forces act in the line joining the elements, the electromotive force on $$ds'$$ is also in this line, and in order to obtain the electromotive force in the direction of $$ds'$$ we must resolve the force in that direction. To apply Weber's formula, we must calculate the various terms which occur in it, on the supposition that the